Chris Moreh, 2023
Week 1
Gamblers, God, Guinness and peas
A brief history of statistics
Gaming chance
1 2 3 4
He threw four knucklebones on to the table and committed his hopes to the throw. If he threw well, particularly if he obtained the image of the goddess herself, no two showing the same number, he adored the goddess, and was in high hopes of gratifying his passion; if he threw badly, as usually happens, and got an unlucky combination, he called down imprecations on all Cnidos, and was as much overcome by grief as if he had suffered some personal loss.
— Lucian of Samosata (c. 125 – 180), writing in his trademark satirical style about a young man who fell in love with Praxiteles’s Aphrodite of Knidos; cited in F. N. David (1955:8)
Suppose you are presented with a large urn full of tiny white and black pebbles, in a ratio that’s unknown to you. You begin selecting pebbles from the urn and recording their colors, black or white. How do you use these results to make a guess about the ratio of pebble colors in the urn as a whole?
Bernoulli’s solution, more technically:
For any given \(\epsilon\) > 0 and any \(s\) > 0, there is a sample size \(n\) such that, with \(w\) being the number of white pebbles counted in the sample and \(f\) being the true fraction of white pebbles in the urn, the probability of \(w/n\) falling between \(f − \epsilon\) and \(f + \epsilon\) is greater than \(1 − s\).
the fraction \(w/n\) is the ratio of white to total pebbles we observe in our sample
\(\epsilon\) (epsilon) captures the fact that we may not see the true urn ratio exactly thanks to random variation in the sample; larger samples help assure that we get closer to the “true” value, but uncertainty always remains
\(s\) reflects just how sure we want to be; for example, set \(s\) = 0.01 and be 99% percent sure.
“moral certainty” as distinct from absolute certainty of the kind logical deduction provides